Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Show Recent Comments

10 Most Recent Blog Comments:

Write_Mikey_Write!

about 19 hours ago
In response to "Global Domination"

A little girl wearing a Hello Kitty shirt in Middle-of-Nowhere, Siberia will certainly make you wonder how it got there. My grandson has a souvenir t-shirt with a Terex 33-19 "Titan" on it. When it was built in 1973, the Titan was the largest, highest payload capacity truck ever built, a title it held for 25 years. The thing is, the truck on that t-shirt is on display in Sparwood, BC, Canada - over 1000 miles from here, and his mom got the shirt at a local thrift store. Makes me wonder.

Oh, yeah - I thought 'Schrödinger's Hello Kitty' was funny, too.

1 Like

Adherennium Plotting something

about 21 hours ago
In response to "Global Domination"

For whatever warped reason, 'Schrödinger's Hello Kitty' has tickled my funny bone.

3 Like

Brandiwyn🎶

Nov 12, 2024 at 10:50am
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

February 22, 2020 at 12:06am

#976062

Mutually Insured Destruction

I try to stay away from overly political issues here, but this one affects all of us here in the US. Other countries, here's your chance to pile on and laugh at us.

https://www.theguardian.com/us-news/2019/nov/14/health-insurance-medical-bankrup...

'I live on the street now': how Americans fall into medical bankruptcy
Having health insurance is often not enough to save Americans from massive debts when serious illness strikes

“I paid my $300 copay. After the surgery, I started receiving all these invoices and came to find out the only thing covered was my bed because the hospital was out of network,” said LeClair. “My bills were hundreds of thousands of dollars, so I had no choice but to file bankruptcy.”

One of the biggest issues I had to face upon early retirement was health care. Or, to be precise, how to pay for it; the nearest hospital is less than two miles from me, and it's one of the best.

Fortunately, my ex and I parted on good terms, and we stayed legally married for several years specifically so I could stay on her health insurance (she worked in a lab associated with said hospital, so there was no "in-network" question).

A good thing, too, because it was during this period that I had a heart attack. I don't remember what the total pre-insurance cost of the ER, treatment, and subsequent hospital stay was, exactly, but there were six numbers before the decimal point. The two after the decimal point were irrelevant. Nor do I remember how much I had to pay out of pocket, because it wasn't a hell of a lot.

When we finalized the divorce, I was able to continue through COBRA, to the tune of about $550 a month.

When he lost his job due to the prolonged hospital stay and leg amputation, his employer offered Cobra, a health insurance program for employees who lose their job or have a reduction in work hours, but he couldn’t afford it.

I paid that premium every month, on time, but the people administering it are complete morons who, moreover, don't give a damn. And when the 18-month extension ran out, I got a notice that the insurance was cancelled, not because time ran out, but, according to their termination notice, for "non-payment."

This happened in a July, and the official "non-payment" declaration meant that I couldn't sign up for Obamacare even if I tried. So there I was, "pre-existing condition" and all, screwed.

When the open enrollment time rolled around - the following December - the cheapest plan I could find, the one with the highest deductible, was about $1500 a month -- and the nearby hospital, the one where all my doctors were, was, you guessed it, out of network. $1500 a month, if you're not math-inclined, is $18,000 a year, which is more than I got paid at my first full-time, salaried job. Which included health insurance as a benefit.

In short, fuck that.

I spent a year and a half uninsured. Not a huge deal in hindsight, because I was fortunate in that I have relatively cheap prescriptions and only three doctor visits a year, with only one of those being a specialist (cardiology). But if I'd fallen seriously ill again, I'd have been boned.

It was only this past December that I found a slightly more affordable plan, with my hospital in network, though it still requires a high deductible.

However, I have recurring nightmares of them deciding to simply not pay for anything that actually happens to me, like the person in the linked article:

At the time, Hillman was receiving several collection notices in the mail for past hospital stays and tests amounting to several thousand dollars, often having no knowledge of the bills that health insurance didn’t cover until receiving the collection notices.

So, you can pay them $800 a month, with a $7500 deductible (for those of you in civilized countries, that's the amount I have to pay out of pocket each year before insurance kicks in -- assuming of course that I get billed for that much, which, to be frank, is easy, because that's like two aspirins and an EKG) with absolutely no guarantee that insurance would actually cover something you need.

“One of the biggest hurdles you face as a patient is just the sheer confusion of the process. You think you just show up and present your card, sometimes pay a copay, and that’s it. You don’t expect all these plan limitations and authorizations,” Hillman added. “What are you going to do if your authorization gets denied? You don’t really have a choice to not go get care. All these processes that are in the finest of fine print. And sometimes it feels like you are literally paying for nothing.”

Oh, but I do have a choice. I can refuse treatment, and possibly die as a result. This might be difficult to implement in practice, though. If I were to collapse on a sidewalk somewhere, it's unlikely that they'll just let me croak there like I want, instead of charging me $100,000 for an ambulance ride to the ER. After all, someone might trip over my corpse and then sue... I dunno, someone.

Thing is, I'd rather die rich than live poor. But I don't know how to communicate that to the paramedics.

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